How do you show that #f(x)=6x^2 - 24x + 22# satisfies the hypotheses of Rolle's theorem on [0,4]?

Answer 1

Use the fact that this function is a polynomial.

#f(x)=6x^2 - 24x + 22# is continuous on #[0,4]# because it is a polynomial and polynomials are continuous at every real number.
#f# is differentiable on #(0,4)# because it is a polynomial so its derivative is a polynomial which is defined for all real numbers, so of course the derivative is defined on #(0,4)#
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Answer 2

To show that ( f(x) = 6x^2 - 24x + 22 ) satisfies the hypotheses of Rolle's theorem on the interval ([0,4]), we need to demonstrate the following:

  1. The function ( f(x) ) is continuous on ([0,4]).
  2. The function ( f(x) ) is differentiable on ( (0,4) ).
  3. ( f(0) = f(4) ).

Let's verify each condition:

  1. Since ( f(x) = 6x^2 - 24x + 22 ) is a polynomial function, it is continuous everywhere, including on the interval ([0,4]).

  2. ( f(x) = 6x^2 - 24x + 22 ) is a polynomial function, which implies it is differentiable everywhere. Therefore, it is differentiable on the interval ( (0,4) ).

  3. ( f(0) = 22 ) and ( f(4) = 6(4)^2 - 24(4) + 22 = 22 ).

Since all three conditions are satisfied, we can conclude that ( f(x) = 6x^2 - 24x + 22 ) satisfies the hypotheses of Rolle's theorem on the interval ([0,4]).

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Answer from HIX Tutor

When evaluating a one-sided limit, you need to be careful when a quantity is approaching zero since its sign is different depending on which way it is approaching zero from. Let us look at some examples.

When evaluating a one-sided limit, you need to be careful when a quantity is approaching zero since its sign is different depending on which way it is approaching zero from. Let us look at some examples.

When evaluating a one-sided limit, you need to be careful when a quantity is approaching zero since its sign is different depending on which way it is approaching zero from. Let us look at some examples.

When evaluating a one-sided limit, you need to be careful when a quantity is approaching zero since its sign is different depending on which way it is approaching zero from. Let us look at some examples.

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